Irregular shock waves formation as continuation of analytic solutions

Abstract

This paper is devoted to the blow up of analytic solutions with the emergence of irregular solutions. At first, we consider the Panov-Shelkovich system where such types of -wave solutions have been explicitly exhibited. We propose a method to reduce this system of nonlinear PDEs to a system of two ODEs in Banach spaces, which permits to obtain theoretical existence of approximate solutions for the Cauchy problem by constructing weak asymptotic solutions. Further, this method allows to study these PDEs through their ODEs representation by basic elementary numerical schemes whose construction is very easy. Then, we observe the expected results from the previous theoretical method; this also gives confidence in the mathematical proofs. We prove that this method gives back the classical analytic solutions using an abstract Cauchy-Kovalevska theorem in scales of Banach spaces. We also study solutions in the form of for other similar systems. Indeed, we show formation of very irregular shock waves when the existence time of a classical analytic solution is over. Finally, we sketch adaptations to provide weak asymptotic solutions to the 3-D Euler-Poisson equations with application to pressureless fluid dynamics and cosmology.

Keywords:

• irregular shock waves
• weak asymptotic methods
• Panov-Shelkovich systems
• partial differential equations
• ordinary differential equations in Banach spaces

AMS Subject Classifications:

• 35A10
• 35A24
• 35D99
• 35L67

Acknowledgements

The author is indebted to the two referees. This work has been done after a suggestion of Vladimir Shelkovich who left us last year.

Notes

Dedicated to the Memory of Vladimir Shelkovich.

This research was supported by the Fundação de Amparo a Pesquisa do Estado de São Paulo, processo 2012/15780-9.